The problem is not the neutron decay. It's that the neutron interacts with your detector.
Consider a very interesting and very difficult decay, like KL --> pi0 nu nubar. That has a SM branching fraction that's very, very tiny: like 10-12. If you want to see the decay - which looks like two photons plus nothing - you need to beat down the background to well under one per trillion kaons. While it's hard for a neutron to fake a photon (or two to fake two), it's not one-in-a-trillion times hard. More like one in few a thousand. So the name of the game is to get the neutrons out of your sample.
This is hard. The least hard thing to do is to take advantage of the fact that neutrons tend to arrive late and interact more slowly, so you place tight timing cuts on the interaction. This works best when you kaons arrive in a small window and most of the beam spill is empty. But a mostly empty beam spill doesn't give you that many kaons. That's why i said there's this tension between beam purity and beam intensity.
KLOE gets a very pure sample by using e+e- annihilation to the phi(1020). The price they paid is that the number of events is measured in millions, so they don't see these ultra-rare decays.